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Unformatted text preview: probably allocated to each of the n ﬂoors, the distinct patterns
are not equally likely (Why? For example, if there are 3 ﬂoors and 2
robots, what is the probability of all robots going to the 1st ﬂoor?
What is the probability of one robot going to the 1st ﬂoor and the
other robot going to the 2nd ﬂoor?). The correct probability should be
nr Problem 6 Problem 6. In an experiment k balls are drawn randomly without
replacement from an urn containing 1 red ball, 2 blue balls and 17
yellow balls. All the balls are identical apart from their colours.
1 Suppose 0 ≤ k ≤ 17. Show how you may obtain the number of distinct
combinations of the k balls by considering the identity
(1 + x )(1 + x + x 2 )(1 − x )−1 ≡ (1 − x 2 )(1 − x 3 )(1 − x )−3 . 2 Show that there are six distinct ways of drawing three balls randomly
without replacement from the urn. Solution to Problem 6 Solution.
1 The number of ways of drawing k balls from the urn is equal to the
coeﬃcient of x k in the expression (why? Note that
(1 + x )(1 + x + x 2 )(1 − x )−1 = (1 + x )(1 + x + x 2 )(1 + x + x 2 + x 3 + ...). 2 The number of ways of drawing k balls from the urn is equal to the
coeﬃcient of x k in the above expression.)
Using the expression on the RHS, we get
(1 − x 2 )(1 − x 3 )(1 − x )−3 = (1 − x 2 − x 3 + x 5 )(1 + 3x + 6x 2 + 10x 3 + ...).
From above, we see that the coeﬃcient of x 3 is −1 − 3 + 10 = 6. Problem 7 Problem 7. (Class test Fall 2008) A lady has three rings and eight
distinguishable ﬁngers (excluding the two thumbs).
1 Suppose the three rings are identical.
1 2 3 2 How many distinct ways are there to wear the three rings on the same
How many distinct ways are there to wear two rings on the same ﬁnger
and the third ring on a diﬀerent ﬁnger?
How many distinct ways are there to wear the three rings on three
separate ﬁngers? Now suppose the three rings are distinguishable. Using part (a) or
otherwise, ﬁnd the total number of distinct ways to wear the three
rings on the lady’s eight ﬁng...
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